Non-Standard Analysis, Multiplication of Schwartz Distributions and Delta-Like Solution of Hopf's Equation

TL;DR

This paper develops a new algebra of generalized functions that embeds Schwartz distributions, enabling multiplication and the construction of delta-like solutions to Hopf's equation, surpassing previous methods in Colombeau's framework.

Contribution

It introduces an improved algebraic framework for multiplying Schwartz distributions and constructs novel delta-like solutions to Hopf's equation within this setting.

Findings

Constructed an algebra of generalized functions $^*\mathcal{E}( ^d)$.

Embedded Schwartz distributions into this algebra.

Proved existence of a weak delta-like solution to Hopf's equation.

Abstract

We construct an algebra of generalized functions $^*\mathcal{E}(\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\mathcal{D}^\prime(\mathbb{R}^d)$ into $^*\mathcal{E}(\mathbb{R}^d)$ and thus present a solution of the problem of multiplication of Schwartz distributions which improves J.F. Colombeau's solution. As an application we prove the existence of a weak delta-like solution in ${^*\mathcal{E}(\mathbb{R}^d)}$ of the Hopf equation. This solution does not have a counterpart in the classical theory of partial differential equations. Our result improves a similar result by M. Radyna obtained in the framework of perturbation theory.